Finite knot surgeries and Heegaard Floer homology

Abstract

It is well known that any $3$–manifold can be obtained by Dehn surgery on a link, but not which ones can be obtained from a knot or which knots can produce them. We investigate these two questions for elliptic Seifert fibered spaces (other than lens spaces) using the Heegaard Floer correction terms or $d$–invariants associated to a $3$–manifold $Y$ and its torsion ${Spin}^c$ structures. For ${\pi }_1\left(Y\right)$ finite and $\left|H_1\left(Y\right)\right|\le 4$, we classify the manifolds which are knot surgery and the knot surgeries which give them; for $\left|H_1\left(Y\right)\right|\le 32$, we classify the manifolds which are surgery and place restrictions on the surgeries which may give them.